\Header{linear.approx}{Linear Approximation of Bootstrap Replicates}
\keyword{nonparametric}{linear.approx}
\begin{Description}\relax
This function takes a bootstrap object and for each bootstrap replicate
it calculates the linear approximation to the statistic of interest for 
that bootstrap sample.\end{Description}
\begin{Usage}
\begin{verbatim}
linear.approx(boot.out, L=NULL, index=1, type=NULL, t0=NULL, 
              t=NULL, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{boot.out}] An object of class \code{"boot"} representing a nonparametric bootstrap.  It will
usually be created by the function \code{boot}.

\item[\code{L}] A vector containing the empirical influence values for the statistic of
interest.  If it is not supplied then \code{L} is calculated through a call 
to \code{empinf}.

\item[\code{index}] The index of the variable of interest within the output of 
\code{boot.out\$statistic}. 

\item[\code{type}] This gives the type of empirical influence values to be calculated.  It is
not used if \code{L} is supplied.  The possible types of empirical influence
values are described in the helpfile for \code{empinf}.

\item[\code{t0}] The observed value of the statistic of interest.  The input value is used only 
if one of \code{t} or \code{L} is also supplied.  The default value is 
\code{boot.out\$t0[index]}.  If \code{t0} is supplied but neither \code{t} nor \code{L} are supplied
then \code{t0} is set to \code{boot.out\$t0[index]} and a warning is generated.

\item[\code{t}] A vector of bootstrap replicates of the statistic of interest.  If \code{t0} is 
missing then \code{t} is not used, otherwise it is used to calculate the empirical 
influence values (if they are not supplied in \code{L}).  

\item[\code{...}] Any extra arguments required by \code{boot.out\$statistic}.  These are needed if
\code{L} is not supplied as they are used by \code{empinf} to calculate empirical 
influence values.

\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The linear approximation to a bootstrap replicate with frequency vector \code{f}
is given by \code{t0 + sum(L * f)/n} in the one sample with an easy extension
to the stratified case.  The frequencies are found by calling \code{boot.array}.\end{Details}
\begin{Value}
A vector of length \code{boot.out\$R} with the linear approximations to the
statistic of interest for each of the bootstrap samples.\end{Value}
\begin{References}\relax
Davison, A.C. and Hinkley, D.V. (1997) 
\emph{Bootstrap Methods and Their Application}. Cambridge University Press.\end{References}
\begin{SeeAlso}\relax
\code{\Link{boot}}, \code{\Link{empinf}}, \code{\Link{control}}\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
# Using the city data let us look at the linear approximation to the 
# ratio statistic and its logarithm. We compare these with the 
# corresponding plots for the bigcity data 

data(city)
data(bigcity)
ratio <- function(d, w)
     sum(d$x * w)/sum(d$u * w)
city.boot <- boot(city, ratio, R=499, stype="w")
bigcity.boot <- boot(bigcity, ratio, R=499, stype="w")
par(pty="s")
par(mfrow=c(2,2))

# The first plot is for the city data ratio statistic.
city.lin1 <- linear.approx(city.boot)
lim <- range(c(city.boot$t,city.lin1))
plot(city.boot$t, city.lin1, xlim=lim,ylim=lim, 
     main="Ratio; n=10", xlab="t*", ylab="tL*")
abline(0,1)

# Now for the log of the ratio statistic for the city data.
city.lin2 <- linear.approx(city.boot,t0=log(city.boot$t0), 
                           t=log(city.boot$t))
lim <- range(c(log(city.boot$t),city.lin2))
plot(log(city.boot$t), city.lin2, xlim=lim, ylim=lim, 
     main="Log(Ratio); n=10", xlab="t*", ylab="tL*")
abline(0,1)

# The ratio statistic for the bigcity data.
bigcity.lin1 <- linear.approx(bigcity.boot)
lim <- range(c(bigcity.boot$t,bigcity.lin1))
plot(bigcity.lin1,bigcity.boot$t, xlim=lim,ylim=lim,
     main="Ratio; n=49", xlab="t*", ylab="tL*")
abline(0,1)

# Finally the log of the ratio statistic for the bigcity data.
bigcity.lin2 <- linear.approx(bigcity.boot,t0=log(bigcity.boot$t0), 
                              t=log(bigcity.boot$t))
lim <- range(c(log(bigcity.boot$t),bigcity.lin2))
plot(bigcity.lin2,log(bigcity.boot$t), xlim=lim,ylim=lim,
     main="Log(Ratio); n=49", xlab="t*", ylab="tL*")
abline(0,1)

par(mfrow=c(1,1))
\end{ExampleCode}
\end{Examples}

